Free particles particle wavepacket

It is not difficult to show that, for a constant potential, equation (A3.11.218) and equation (A3.11.219) can be solved to give the free particle wavepacket in equation (A3.11.7). More generally, one can solve equation (A3.11.218) and equation (A3.11.219) numerically for any potential, even potentials that are not quadratic, but the solution obtained will be exact only for potentials that are constant, linear or quadratic. The deviation between the exact and Gaussian wavepacket solutions for other potentials depends on how close they are to bemg locally quadratic, which means... 

Time evolution of a one-dimensional free particle wavepacket... 

We refer to the wavefunction (2.98) in the present context as a wavepacket It is a local function in position space that can obviously be expanded in the complete set of free particle waves (2.82). Below we see that this function is also localized in momentum space. 

Another theory which is used to describe scattering problems and which blends together classical and quantum mechanics is the semiclassical wavepacket approach [53]. The basic procedure comes from the fact that wavepackets which are initially Gaussian remain Gaussian as a function of time for potentials that are constant, linear or quadratic functions of the coordinates. In addition, the centres of such wavepackets evolve in time in accord with classical mechanics. We have already seen one example of this with the free particle wavepacket of equation (A3.11.7). Consider the general quadratic Hamiltonian (still in one dimension but the generalization to many dimensions is straightforward)... 

The wave function for a given electro-nuclear state is the product of the universal electronic function, the relative vibrational motion of the nuclei, the rotational wave function and the center-of-mass free particle wavepacket. 

To form a photon-like particle, the elementary normal EMS modes now have to be superimposed to create a wavepacket of finite axial extensions and of finite linewidth in wavelength space. Here we are free to choose an amplitude factor Go of the generating function (90) having the form... 

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